Books in graphs


A set of q triangles sharing a common edge is called a book of size q. We write β(n,m) for the maximal q such that every graph G(n,m) contains a book of size q. In this note (1) we compute β(n, cn2) for infinitely many values of c with 1/4 < c < 1/3, (2) we show that if m ≥ (1/4-α)n2 with 0 < α < 17-3, and G has no book of size at least (1/6-2α1/3)n then G contains an induced bipartite graph G1 of order at least (1-α1/3)n and minimal degree δ(G1)≥(1/2 - 4α1/3)n, (3) we apply the latter result to answer two questions of Erdos concerning the booksize of graphs G(n,n2/4-f(n)n) every edge of which is contained in a triangle, and 0

Publication Title

European Journal of Combinatorics